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─次元周期ポテンシャル中の電子のエネルギ-と波動関数の視覚的理解

照彰 皆川

Published Sep 30, 1996• 0 citations
OpenAlex

Abstract

The wave function, the energy, the kinetic energy and the potential energy of an electron moving in one-dimensional periodic potentials were obtained by calculating Schrodinger equation numerically with a personal computer. It was shown that we can visually and intuitively understand the variations of energy, kinetic energy and potential energy by comparing the shape of wave function obtained with the shape of periodic potential. It was also indicated that many other things are understood by diagraming the wave function and the periodic potential.|一次元周期ポテンシャルの中を運動する電子のエネルギー, 運動エネルギー, ポテンシャルエネルギー及び波動関数がパーソナルコンピュータを用いたシュレーディンガー方程式の数値計算によってもとめられた。求められた波動関数の形とポテンシャルの形とを図示し, 比べることによって, 電子のエネルギー, 運動エネルギー及びポテンシャルエネルギーの増減を視覚的かつ直感的に理解できることが示された。波動関数とポテンシャルを図示し, 視覚化することが, 多くの現象を理解するのに非常に有効であることが多くの例で示されている。

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The wave function, the energy, the kinetic energy and the potential energy of an electron moving in one-dimensional periodic potentials were obtained by calculating Schrodinger equation numerically with a personal computer. It was shown that we can visually and intuitively understand the variations of energy, kinetic energy and potential energy by comparing the shape of wave function obtained with the shape of periodic potential. It was also indicated that many other things are understood by diagraming the wave function and the periodic potential.|一次元周期ポテンシャルの中を運動する電子のエネルギー, 運動エネルギー, ポテンシャルエネルギー及び波動関数がパーソナルコンピュータを用いたシュレーディンガー方程式の数値計算によってもとめられた。求められた波動関数の形とポテンシャルの形とを図示し, 比べることによって, 電子のエネルギー, 運動エネルギー及びポテンシャルエネルギーの増減を視覚的かつ直感的に理解できることが示された。波動関数とポテンシャルを図示し, 視覚化することが, 多くの現象を理解するのに非常に有効であることが多くの例で示されている。
The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .
We found a simple procedure for the solution of the time - independent Schrodinger equation in one dimension without making any approximation. The wave functions are always periodic. Two difficulties may be encountered: one is to solve the equation E=U(x), where E and U(x) are the total and potential energies, respectively, and the other is to calculate the integral of the square root of U(x). If these calculations cannot be made analytically, it should then be performed by numerical methods. To find the energy and the wave function of the ground state, there is no need to calculate this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(x).
2重障壁ポテンシャルに障壁よりも低いエネルギーをもつ台形波束を入射して共鳴トンネル現象が観測されるか調べた。時間を含む1次元シュレディンガー方程式を数値的に解き,パソコン画面上で波束の運動を観察した。共鳴は,2重障壁の間に波束が長時間にわたって多量にたまることによって識別できた。また,波束をフーリエ変換して得た波数分布から透過波と反射波の割合をみることによって,波束が2重障壁をトンネル透過する程度が判別できた。2重障壁の固有の量子エネルギー準位に対応して台形波束が2重障壁を共鳴的にトンネル透過することが,いろいろな条件の場合について,鮮明に観測された。その結果,2重障壁は特定のエネルギーを持つ波束すなわち粒子のみを選択的に透過させる優れたフィルターの役割をすることがわかった。|A resonance tunnel effect was investigated by leading a wave-packet of trapezoid shape into double potential barriers. The time-dependent Schrödinger equation was numerically solved, and the motion of a wave packet was observed on the screen of a personal computer. The state that a wave packet is resonant with double potential barriers showed a distinct phenomenon that a large part of a wave packet is stored between two barriers for a long time. Tunneling of a wave packet was clearly recognized by comparing an amount of the transmitted wave with that of the reflected one in the distribution of wave vectors that was obtained by the Fourier transformation of a wave packet. A resonance tunnel effect was demonstrated for various kinds of double potential barriers. It was showed that a couple of double potential barriers functions as an excellent filter that allows particles with specified energies to pass through.
The Schrodinger equation is an equation that describes the properties of a wave. The Schrodinger equation is solved using a variable separation method that produces a wave function. The wave function using the variable separation method is divided into 2 parts, namely the radial part and the angular part. The energy spectrum states the energy levels of electrons. Energy spectrum and wave function are related to each other. The energy level in the Schrodinger wave function in the form of kinetic energy is the energy for electrons to move from one point to another. The research aims to describe the kinetic energy spectrum of electrons in the radial wave function. The type of research used is basic research, namely the development of existing theories. The result obtained is a radial wave function with a certain quantum number n and an integrated Laplace constant that can produce electron kinetic energy, namely by integrating the wave function with each quantum number. The kinetic energy has its own value according to the quantum number. Kinetic energy in quantum numbers (n = 1,l = 0, Ek =171.18 eV ; n =2,l = 0, Ek =399.09 eV n =2,l = 1, Ek =152.16 eV n =3,l = 0, Ek =5096.81 eV n =3,l = 1, Ek =81142.43 eV n =3,l = 2, Ek =21.979 eV). The conclusion of this research is that the kinetic energy spectrum of electrons produces a directly proportional relationship to the quantum number n, where the greater the quantum number n, the greater the spectrum or kinetic energy level of the electron.
The propagation of one dimensional electron wave function in periodic potential have been simulated. Gaussian wave function was chose to simulate the wave packet for the electron. The simulation or periodic potential shows that it follows Bloch theorem. Many problems related with the shape of potential are simulated also. Initial energy chose as tested parameter, which correspond to the speed of wave packet and related with the energy of the electron, and its show that for greater energy the spreading of wave packet become slower. The dynamical parameter makes possibilities to evaluate the propagation every time we need.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.